Closed range composition operators on Hilbert function spaces

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.1 (1-26)

J. Math. Anal. Appl. ••• (••••) •••–•••

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Closed range composition operators on Hilbert function spaces Pratibha Ghatage a , Maria Tjani b,∗ a b

Department of Mathematics, Cleveland State University, Cleveland, OH 44115, USA Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA

a r t i c l e

i n f o

Article history: Received 24 December 2014 Available online xxxx Submitted by J. Bonet Keywords: Composition operator Hilbert function space Bergman space Hardy space Dirichlet space Closed range

a b s t r a c t We show that a Carleson measure satisfies the reverse Carleson condition if and only if its Berezin symbol is bounded below on the unit disk D. We provide new necessary and sufficient conditions for the composition operator to have closed range on the Bergman space. The pull-back measure of area measure on D plays an important role. We also give a new proof in the case of the Hardy space and conjecture a condition in the case of the Dirichlet space. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Let ϕ be an analytic self-map of the unit disk D. The composition operator with symbol ϕ is defined by Cϕ (f ) = f ◦ ϕ , for any function f that is analytic on D. Littlewood in 1925 proved a subordination principle, which in operator theory language, says that composition operators are bounded in the Hardy space H 2 , the Hilbert space of analytic functions on D with square summable power series coefficients. This is the first setting by which many properties of the composition operator such as boundedness, compactness, and closed range have been studied. It is natural to study these properties on other function spaces. Let H be a Hilbert space of analytic functions on D with inner product · , ·. We say that H is a Hilbert function space if all point evaluations are bounded linear functionals. By the Riesz representation theorem, for each z ∈ D, there exists a unique element Kz of H, called the reproducing kernel at z, such that for each f ∈ H, f (z) = f, Kz . For each z ∈ D we have |f (z)| ≤ f  Kz  = f  Kz (z)1/2 , * Corresponding author. E-mail addresses: [email protected] (P. Ghatage), [email protected] (M. Tjani). http://dx.doi.org/10.1016/j.jmaa.2015.06.009 0022-247X/© 2015 Elsevier Inc. All rights reserved.

(1)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.2 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

2

where  ·  denotes the norm in H. In particular for each z, w ∈ D we have | < Kw , Kz > | = |Kw (z)| ≤ Kw (w)1/2 Kz (z)1/2 .

(2)

For each w ∈ D, the normalized reproducing kernel in H is kw (z) =

Kw (z) , ||Kw ||

z ∈D.

(3)

If Cϕ is a bounded operator on H, then by Theorem 1.4 in [11] we have Cϕ∗ (Kz ) = Kϕ(z)

(4)

||Kϕ(z) || ≤ ||Cϕ || ||Kz || .

(5)

and hence

Cima, Thomson and Wogen in [9] were the first to study closed range composition operators in H 2 . Their results were in terms of the boundary behavior of ϕ. Next, Zorboska in [27] studied closed range composition operators in H 2 and in the weighted Hilbert Bergman space in terms of properties of ϕ inside D. Since then several authors studied this problem in different Banach spaces of analytic functions, see for example [14, 2–4,18,25]. In this paper we continue the study of closed range composition operators on the Bergman space A2 , the Hardy space H 2 and the Dirichlet space D. We will define and discuss properties of these spaces as well as other preliminary work in Section 2. In Section 3 we develop general machinery that can be useful in studying closed range composition operators in any Hilbert function space H with reproducing kernel Kz . Given ε > 0, let   Λε = Λε (H) = z ∈ D : ||Kϕ(z) || > ε||Kz || and Gε = Gε (H) = ϕ(Λε ). In Section 3 we give a necessary condition for Gε to intersect each pseudohyperbolic disk in D, see Proposition 3.2. It is useful in our results in Section 4 and in Section 6. In Section 4 we build on known results about closed range composition operators on A2 provided in [2] and [25]. It is well known that μ is a Carleson measure on D if and only if the Berezin symbol of μ is bounded; our first main result in Section 4 is an analog of this for Carleson measures that satisfy the reverse Carleson condition, see Theorem 4.1. We use this to provide necessary and sufficient conditions for the pull-back measure of area measure on D to satisfy the reverse Carleson condition, see Theorems 4.2 and 4.3. Akeroyd and Ghatage in [2] showed that Cϕ is closed range in H = A2 if and only if for some ε > 0 the set Gε above satisfies the reverse Carleson condition. This means that Gε intersects every pseudohyperbolic disk, of some fixed radius r ∈ (0, 1), in a set that has area comparable to the area of each pseudohyperbolic disk. In Theorem 4.4 we show that in fact this is equivalent to Gε merely having non-empty intersection with each pseudohyperbolic disk. Moreover we provide an analog of [9, Theorem 2] in A2 in terms of the pull-back measure of normalized area measure on D; we also show that Cϕ is closed range on A2 if and only if for all w ∈ D, ||Cϕ Kw ||  ||Kw ||. Lastly we provide a condition that makes it easy to check whether Cϕ is closed range on A2 , see (e) of Theorem 4.4. In Section 5 we revisit closed range composition operators on H 2 . The first main result of this section is not new. It is a combination of [25, Theorem 5.4] and a result in Luery’s thesis [18, Theorem 5.2.1]. Our new short proof uses pseudohyperbolic disks. Zorboska proved in [27] that for univalent symbols, Cϕ is closed range on A2 if and only if it is closed range on H 2 . We extend Zorboska’s result to include all symbols ϕ ∈ D. Akeroyd and Ghatage show in [2] that the only univalent symbols that give rise to a closed range Cϕ

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.3 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

3

on A2 are given by conformal automorphisms. We provide an example of a closed range Cϕ on H 2 and on A2 where the symbol ϕ is an outer function. Note that all other known such examples involve inner functions. In Section 6 we apply our techniques to the Dirichlet space. Nevanlinna type counting functions in A2 and in H 2 were instrumental in determining closed range. In comparison Luecking showed in [17] that the Dirichlet space analog does not determine closed range. We provide a new Nevanlinna type counting function in D and introduce three Carleson measures and study their properties, see Propositions 6.1 and 6.2 and Corollary 6.1. We end the paper with a conjecture about closed range composition operators in D. Throughout this paper C denotes a positive and finite constant which may change from one occurrence to the next but will not depend on the functions involved. Given two quantities A = A(z) and B = B(z), z ∈ D, we say that A is equivalent to B and write A  B if, C A ≤ B ≤ C A. 2. Preliminaries For each p ∈ D let αp denote the Mobius transformation exchanging 0 and p, αp (z) =

p−z , 1 − pz

(6)

and Aut(D) denote the set of all Mobius transformations of D. The pseudo-hyperbolic distance ρ between two points w, z in D defined by ρ(z, w) = |αz (w)| = |

z−w |, 1 − wz

(7)

is Mobius invariant, that is for all z, w ∈ D and γ ∈ Aut(D), ρ(γ(z), γ(w)) = ρ(z, w) , and satisfies a strong form of the triangle inequality; given z, w, ζ ∈ D, ρ(z, w) ≤

ρ(z, ζ) + ρ(ζ, w) . 1 + ρ(z, ζ)ρ(ζ, w)

(8)

Moreover it has the following important property: 1 − ρ(z, w)2 =

(1 − |z|2 )(1 − |w|2 ) . |1 − wz|2

(9)

The pseudo-hyperbolic disk D(z, r) centered at z with radius r ∈ (0, 1) is D(z, r) = {w : ρ(w, z) < r} .

(10)

x+y As mentioned in [12, page 39], the function f (x, y) = 1+xy attains a maximum value in the rectangle [0, r] × [0, s] at the point (r, s). It now follows by (8) that if z ∈ D(ζ, r) and w ∈ D(ζ, s) then

ρ(z, w) ≤

r+s . 1 + rs

(11)

Let A denote area measure on D normalized by the condition A(D) = 1. By [26, Proposition 4.5] if r ∈ (0, 1) is fixed and z ∈ D(z0 , r) then A(D(z, r))  (1 − |z|2 )2  A(D(z0 , r))  (1 − |z0 |2 )2 ,

(12)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.4 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

4

and |1 − z 0 z|  1 − |z|2  1 − |z0 |2 .

(13)

The Bergman space A2 is the Hilbert space of analytic functions f on D that are square-integrable with respect to the area measure A that is,  2 ||f ||A2 = |f (z)|2 dA(z) < ∞ . (14) D

By [26, Theorem 4.28] an equivalent norm on A2 is given by 

(1 − |z|2 )2 |f  (z)|2 dA(z) .

||f ||2A2  |f (0)|2 +

(15)

D

As was mentioned in the Introduction, the Hardy space H 2 is the Hilbert space of analytic functions on D   an z n ∈ H 2 then ||f ||2H 2 = |an |2 and with square summable power series coefficients. Moreover if f = 2 by the Littlewood–Paley identity an equivalent norm in H is given by  ||f ||2H 2 = |f (0)|2 + (1 − |z|2 )|f  (z)|2 dA(z) . (16) D

The Dirichlet space D is the space of analytic functions on D such that  ||f ||2D = |f (0)|2 + |f  (z)|2 dA(z) < ∞ . D

By [1, Proposition 2.18] the reproducing kernel in the Bergman space A2 is Kw (z, A2 ) := Kw (z) =

1 and ||Kw || = (1 − |w|2 )−1 ; (1 − wz)2

(17)

similarly the reproducing kernel in the Hardy space H 2 is Kw (z, H 2 ) := Kw (z) =

1 and ||Kw ||2 = (1 − |w|2 )−1 ; 1 − wz

(18)

the reproducing kernel of D0 = {f ∈ D : f (0) = 0} is Kw (z, D0 ) := Kw (z) = log

1 1 and ||Kw ||2 = log 1 − wz 1 − |w|2

(19)

and the reproducing kernel of D is Kw (z, D) := Kw (z) = 1 + log

1 1 and ||Kw ||2 = 1 + log . 1 − wz 1 − |w|2

(20)

3. On Hilbert function spaces Let H be a Hilbert function space with reproducing kernels Kw , w ∈ D. For ε > 0, let   Λε = Λε (H) = z ∈ D : ||Kϕ(z) || > ε||Kz ||

(21)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.5 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

5

and Gε = Gε (H) = ϕ(Λε ) .

(22)

  Λε (A2 ) = Λε = z ∈ D : 1 − |z|2 > ε (1 − |ϕ(z)|2 ) ,

(23)

By (17) if H = A2 ,

and by (18) if H = H 2 then Λε (H 2 ) = Λε2 (A2 ). The sets Gε (A2 ) were used in [2,3] and in [25] to study closed range composition operators on the Bergman space. Moreover by (19) if H = D0 , then 2

Λε = {z ∈ D : (1 − |z|2 )ε > 1 − |ϕ(z)|2 } .

(24)

Definition 3.1. We say that H ⊆ D is hyperbolically dense if there exists r ∈ (0, 1) such that every point of D is within pseudo-hyperbolic distance r of H. For r ∈ (0, 1) and w ∈ D we define Δw = Δw (r) = ϕ−1 (D(w, r)) .

(25)

Then for ε > 0, the set Gε , defined in (22), is hyperbolically dense if and only if there exists r ∈ (0, 1) such that for each w ∈ D, Δw (r) ∩ Λε = ∅. Definition 3.2. We say that the reproducing kernels satisfy the nearness property if for each r ∈ (0, 1) there exist δ > 0 and cr > 0 such that for each w, ζ ∈ D with |w| ≥ cr , |ζ| ≥ cr and ρ(ζ, w) ≤ r, we have that | < Kζ , Kw > | ≥ δ ||Kw ||.||Kζ || and ||Kw ||  ||Kζ || .

(26)

Next we show that nearness property is quite common. Proposition 3.1. The reproducing kernel functions for the Hardy space H 2 , the Bergman space A2 , the Dirichlet space D and of D0 satisfy the nearness property. Proof. By (9), given r ∈ (0, 1), ρ(z, w) ≤ r if and only if 1 1 − r2 ≤ . (1 − |z|2 ) (1 − |w|2 ) |1 − wz|2

(27)

Therefore by (17) and (18) it is immediate that the reproducing kernels of A2 and H 2 satisfy the nearness property. Next, by taking logarithms of both sides of (27) and using the inequality of arithmetic and geometric means we see that, given r ∈ (0, 1), if ρ(z, w) ≤ r then √

 2

 1 log 1 − |z|2

log

1 1 + log(1 − r2 ) ≤ 2 log 1 − |w|2 |1 − wz|

and letting Kz denote the reproducing kernel in D0 we obtain √

1 log 1−r | < Kw , K z > | 2 2 − ≤ . 2 2||Kw || ||Kz || ||Kw ||Kz ||

(28)

JID:YJMAA 6

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.6 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

 By (19), if |w| > (e − 1)/e then ||Kw || > 1 and by (28) we may find δ > 0 and cr > 0 such that if |z| > cr and |w| > cr then √ δ≤

1 | < K w , Kz > | 2 log 1−r2 − ≤ , 2 2||Kz || ||Kw ||Kz ||

(29)

and the reproducing kernels for D0 and hence of D satisfy the nearness property as well. 2 The proof above can be easily modified to show that the reproducing kernel functions in all weighted Bergman spaces satisfy the nearness property. In the remainder of this section we will assume that H contains all constant functions, therefore by (1), for all z ∈ D we have ||Kz || > 0. Moreover we will assume that H satisfies the following property. (S) The map z → Kz is continuous on D. It is easy to see that property (S) is valid on H 2 , A2 , D0 and D. Below we provide a necessary condition for Gε to be hyperbolically dense. Proposition 3.2. Let H be a Hilbert function space satisfying condition (S) and containing all constant functions. Suppose that Cϕ is a bounded operator on H and that the reproducing kernels of H satisfy the nearness property. If there exists ε > 0 such that Gε is hyperbolically dense, then for every w ∈ D, ||Cϕ Kw ||  ||Kw ||  sup | < Cϕ Kw , z∈D

Kz >|. ||Kz ||

(30)

Proof. Fix ε > 0 such that Gε is hyperbolically dense. Then there exists r ∈ (0, 1) such that given w ∈ D, we may choose zw ∈ Λε such that ρ(w, ϕ(zw )) ≤ r < 1. By the Cauchy–Schwarz inequality and (4) we obtain ||Cϕ Kw || ≥ sup | < Cϕ Kw , z∈D

≥ | < Cϕ Kw ,

Kz >| ||Kz ||

Kzw >| ||Kzw ||

=

| < Kw , Cϕ∗ Kzw > | ||Kzw ||

=

| < Kw , Kϕ(zw ) > | . ||Kzw ||

(31)

By (5) and (21), ||Kzw ||  ||Kϕ(zw ) ||. Therefore by (31), the nearness property of the reproducing kernels of H and since Cϕ is a bounded operator, there exists cr > 0 such that if |w| > cr then C1 ||Kw || ≥ ||Cϕ Kw || ≥ C

||Kw |||.||Kϕ(zw ) ||  ||Kw || . ||Kzw ||

(32)

Moreover by (31) and (32), if |w| ≥ cr C1 ||Kw || ≥ sup | < Cϕ Kw , z∈D

Kz > | ≥ C2 ||Kw || ||Kz ||

(33)

and (30) is valid if |w| ≥ cr . Next suppose that there exists a sequence (wn ) ∈ D with |wn | ≤ cr such that ||Cϕ Kwn || →0. ||Kwn ||

(34)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.7 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

7

Without loss of generality we may assume that there exists |w0 | < cr such that wn → w0 . By assumption H satisfies condition (S). Then ||Kwn || → ||Kw0 || and ||Cϕ Kwn || → ||Cϕ Kw0 ||. By (34) we conclude that, Cϕ Kw0 = 0 and so Kw0 = 0. This is a contradiction, therefore if |w| ≤ cr the first set of inequalities in (30) follows. Lastly, suppose that there exists a sequence (un ) ∈ D with |un | ≤ cr such that Kz supz∈D | < Cϕ Kun , ||K >| z ||

||Kun ||

→0.

(35)

As before, without loss of generality we may assume that there exists |u| < cr such that un → u and by (35) we have that for all z ∈ D, Cϕ Ku (z) = 0 and so Ku = 0. This is a contradiction, therefore if |w| ≤ cr the second set of inequalities in (30) follows as well. 2 Remark 3.1. A close examination of the proof of Proposition 3.2 above reveals that under its hypotheses, there exists r ∈ (0, 1) such that if w ∈ D is outside a certain compact neighborhood of 0, then for each z ∈ Δw (r) ∩ Λε ||Kw ||  ||Cϕ Kw ||  | < Cϕ Kw ,

Kz >|. ||Kz ||

(36)

4. On Bergman spaces Carleson measures play an important role in determining when a composition operator Cϕ is bounded or compact in several Banach spaces of analytic functions. We shall see that they are also important in determining when Cϕ is closed range. Definition 4.1. Let μ be a finite positive Borel measure on D. We say that μ is a (Bergman space) Carleson measure on D if there exists c > 0 such that for all f ∈ A2 

 |f (z)| dμ(z) ≤ c

|f (z)|2 dA(z).

2

D

D

By [26, Theorem 7.4], given 0 < r < 1, μ is a Carleson measure if and only if there exists cr > 0 such that for all w ∈ D, μ(D(w, r)) ≤ cr A(D(w, r)) . By (17), the normalized reproducing kernel in A2 kw (z) =

1 − |w|2 . (1 − wz)2

Let 0 < r < 1, w ∈ D. Then by making a change of variables 



 |αw (z)|2 dA(z)

|kw (z)|2 dA(z) = D\D(w,r)

D\D(w,r)



=1−

 |αw (z)|2 dA(z)

D(w,r)

= 1 − |D(0, r)| .

(37)

JID:YJMAA

AID:19562 /FLA

8

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.8 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

We conclude that  lim sup

r→1 w∈D D\D(w,r)

|kw (z)|2 dA(z) = 0 .

(38)

In fact below we show that this is valid for any positive Carleson measure. We provide a more general version that is crucial in the proof of Theorem 5.1. Proposition 4.1. Let μ be a positive Carleson measure on D, and α, β be such that α + β = 2. Then  lim sup

r→1 w∈D D\D(w,r)

(1 − |w|2 )α (1 − |ζ|2 )β dμ(ζ) = 0 . |1 − wζ|4

(39)

Proof. For a fixed 0 < ρ < 1 we can cover D with pseudohyperbolic disks of radius ρ that do not intersect too often. In fact as shown in [6, Lemma 3.5] there exist (sn ) ⊂ D, and a positive integer M that depends only on ρ, such that D = ∪∞ n=1 D(sn , ρ) and each ζ ∈ D is in at most M of the pseudohyperbolic disks D(an , (ρ + 1)/2), n ∈ N. Fix r ∈ (0, 1) and w ∈ D. Similarly to the above, with ρ = 1/2, there exist (wn ) ⊂ D such that 1 D \ D(w, r) = ∪∞ n=1 D(wn , ) . 2

(40)

Moreover similarly to the argument in [6, Lemma 3.5], for each n, m ∈ N and each ζ ∈ D |αζ (wn ) − αζ (wm ))| ≥ [1 − (ρ + 1)2 /4] ρ/3 = 7/96 . We conclude that there exists a positive integer M , independent of r ∈ (0, 1) and w ∈ D, such that each ζ ∈ D \ D(w, r) is in at most M of the pseudohyperbolic disks D(wn , 3/4), n ∈ N. For each w ∈ D let δn (w) =

(1 − |w|2 )α (1 − |z|2 )β . |1 − wz|4 z∈D(wn , 1 ) sup

(41)

2

Then by [6, Lemma 3.4], or [26, Proposition 4.13], applied to the function constant C such that 

3 δn (w)|D(wn , )| ≤ C 4

D(wn , 34 )

(1 − |w|2 )α (1 − |ζ|2 )β dA(ζ) . |1 − wζ|4

Let  I= D\D(w,r)

(1 − |w|2 )α (1 − |ζ|2 )β dμ(ζ) . |1 − wζ|4

By (40) and (41) and since μ is a Carleson measure I≤

∞ 



n=1 D(wn , 12 )

1−|w|2 (1−wz)2

(1 − |w|2 )α (1 − |ζ|2 )β dμ(ζ) |1 − wζ|4

and by (12), there is a

(42)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.9 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

≤ ≤ 

9

∞ 

1 δn (w)μ(D(wn , )) 2 n=1 ∞ 

3 μ(D(wn , 12 )) δn (w) |D(wn , )| 4 |D(wn , 12 )| n=1 ∞ 

3 δn (w) |D(wn , )| . 4 n=1

Next, by (12) and (42) I≤C



∞ 

n=1 D(wn , 34 )



∞ 



n=1 D(wn , 34 )

(1 − |w|2 )α (1 − |ζ|2 )β dA(ζ) |1 − wζ|4

(1 − |w|2 )2 dA(ζ) . |1 − wζ|4

(43)

Lastly, as shown above, each ζ ∈ D \ D(w, r) is in at most M of the pseudohyperbolic disks D(wn , 3/4), n ∈ N. Therefore  (1 − |w|2 )2 I ≤ C dA(ζ) |1 − wζ|4 D\D(w,r)

for each 0 < r < 1 and w ∈ D. By (38) the result now is clear. 2 Definition 4.2. Let μ be a finite positive Carleson measure on D. We say that μ satisfies the reverse Carleson condition if there exists r ∈ (0, 1) such that for all w ∈ D, A(D(w, r))  μ(D(w, r)) .

(44)

We say that a set G ⊂ D satisfies the reverse Carleson condition if the Carleson measure χG (z) dA(z) satisfies the reverse Carleson condition; Luecking in [15] showed that this is equivalent to 

 |f (z)| dA(z) ≤ C

|f (z)|2 dA(z),

2

D

(45)

G

for all f ∈ A2 . Moreover he showed that Carleson type squares can be used in place of the pseudohyperbolic disks. But proofs are easier using pseudohyperbolic disks. Let μ be a finite positive measure. The Berezin symbol of μ is  (46) μ ˜(w) = |kw (z)|2 dμ(z) , w ∈ D . D

It has played an important role in determining properties of Toeplitz operators and composition operators such as boundedness and compactness, see for example [7]. We shall see below that it is important in questions of closed range of composition operators as well. It is well known that μ is a Carleson measure if and only if the Berezin symbol of μ is bounded above, see for example [26, Theorem 7.5]. Below we show that the reverse Carleson measure analog of this holds as well.

JID:YJMAA 10

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.10 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

Theorem 4.1. A Carleson measure μ satisfies the reverse Carleson condition if and only if μ ˜ is bounded above and below on D. Proof. First assume that the Carleson measure μ satisfies the reverse Carleson condition. By (12) and (13), there exists r ∈ (0, 1), for each w ∈ D we obtain,  C≥ D



(1 − |w|2 )2 dμ(z) ≥ |1 − wz|4

(1 − |w|2 )2 dμ(z) |1 − wz|4

D(w,r)



μ(D(w, r)) A(D(w, r))

1 and therefore μ ˜ is bounded above and below on D. Next assume that μ ˜ is bounded above and below on D. By Proposition 4.1  (1 − |w|2 )2 lim sup dμ(z) = 0 . r→1 w∈D |1 − wz|4

(47)

(48)

D\D(w,r)

We conclude, by (12) and (13), that there exists δ > 0 such that if r > 1 − δ and w ∈ D, μ(D(w, r)) A(D(w, r))  (1 − |w|2 )2  dμ(z) |1 − wz|4

C≥

D(w,r)



≥ D

1 (1 − |w|2 )2 dμ(z) − inf μ ˜(w) 4 |1 − wz| 2 w∈D

=μ ˜(w) − ≥

1 inf μ ˜(w) 2 w∈D

1 inf μ ˜(w) ≥ C 2 w∈D

(49)

and the conclusion follows. 2 The pull-back measure of normalized area measure in D under the map ϕ is defined on Borel subsets of D by μϕ (E) = A(ϕ−1 (E)) .

(50)

˜ϕ is, The Berezin symbol of μ 

 |kw (z)| dμϕ (z) =

|kw (ϕ(z))|2 dA(z) = ||Cϕ kw ||2 .

2

μ ˜ϕ (w) = D

(51)

D

Therefore since Cϕ is a bounded operator on A2 (see for example [24, Section 1.4, Exercise 5]), μ ˜ϕ (w) ≤ ||Cϕ ||2 ||kw ||2 = ||Cϕ ||2 . We conclude that μ ˜ϕ is a bounded function on D, and μϕ is a Carleson measure on D.

(52)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.11 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

Proposition 4.2. The following are equivalent: (a) The Carleson measure μϕ satisfies the reverse Carleson condition. (b)There exists 0 < r < 1 such that for all w ∈ D,  |Cϕ Kw (z)| dA(z)  1 .

11

(53)

Δw (r)

(c) There exists 0 < r < 1 such that for all w ∈ D,  2 2 |Cϕ Kw (z)|2 dA(z)  1 . (1 − |w| )

(54)

Δw (r)

Proof. By (9) and for any w ∈ D, the absolute value of the normalized reproducing kernel in A2 is |Kw (z)| =

1 1 − ρ(z, w)2 , = 2 |1 − wz| (1 − |z|2 ) (1 − |w|2 )

(55)

1 1 − ρ(z, w)2  . A(D(w, r) A(D(w, r)

(56)

and by (12), if z ∈ D(w, r) then |Kw (z)| 

Therefore by (12) and (25), if n = 1, 2 then μϕ satisfies the reverse Carleson condition if and only if there exists 0 < r < 1 such that for w ∈ D,   |Cϕ Kw (z)|n dA(z) = |Kw (z)|n dμϕ (z) Δw (r)

D(w,r)

μϕ (D(w, r)) A(D(w, r))n 1  (1 − |w|2 )2n−2



(57)

and the result follows. 2 The result below is an immediate corollary of Theorem 4.1 applied to the Carleson measure μϕ . ˜ϕ is bounded Theorem 4.2. The Carleson measure μϕ satisfies the reverse Carleson condition if and only if μ above and below on D. Equivalently, for all w ∈ D, ||Cϕ Kw ||  ||Kw || .

(58)

In the theorem below we provide another equivalent condition for μϕ to satisfy the reverse Carleson condition that is essential in the proof of our main result of this section. We single it out so that later we can make clear the analogy between μϕ and the pull-back measure of Lebesgue measure on the unit circle T used in [9]. Theorem 4.3. The Carleson measure μϕ satisfies the reverse Carleson condition if and only if there exists C > 0 such that for all w ∈ D, sup(1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))| ≥ C . z∈D

(59)

JID:YJMAA 12

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.12 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

Proof. First assume that μϕ satisfies the reverse Carleson condition. By Proposition 4.2 there exist 0 < r < 1 and C > 0 such that for all w ∈ D  |Cϕ Kw (z)| dA(z) ≥ C . (60) Δw (r)

We claim that there exist ε > 0 and C  > 0 such that for all w ∈ D  |Cϕ Kw (z)| dA(z) ≥ C  .

(61)

Δw (r)∩Λε

Suppose that (61) is not valid. Then there exists a sequence wn ∈ D such that  lim

n→∞ Δwn (r) ∩Λ1/n

|Cϕ Kwn (z)| dA(z) = 0 .

(62)

Moreover by Theorem 4.2, (12) and (4), for each n ∈ N, 

 |Cϕ Kwn (z)| dA(z) ≤ C Δwn (r) \Λ1/n

||Kϕ(z) || ||Kwn || dA(z) Δwn (r) \Λ1/n

≤C



1 n

||Kz || ||Kwn || dA(z) Δwn (r)

1  n



||Cϕ Kz || ||Kwn || dA(z) Δwn (r)



1 n



||Kz || ||Kwn || dμϕ (z) D(wn ,r)

1 μϕ (D(wn , r)) n A(D(wn , r)) 1  n



and  lim

n→∞ Δwn (r) \Λ1/n

|Cϕ Kwn (z)| dA(z) = 0 .

(63)

Therefore (62) and (63) together contradict (60) and we conclude that (61) is valid. Then by (12) and since μϕ satisfies the reverse Carleson condition, C ≤

sup

(1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))|

A(Δw (r) ∩ Λε ) (1 − |z|2 ) (1 − |w|2 )

(1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))|

A(Δw (r) ∩ Λε ) (1 − |z|2 ) (1 − |ϕ(z)|2 )

z∈Δw (r)∩Λε



sup z∈Δw (r)∩Λε

≤

1 ε

sup z∈Δw (r)∩Λε

(1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))|

μϕ (D(w, r) ∩ Gε ) (1 − |ϕ(z)|2 )2

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.13 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

≤

1 ε

≤C

sup (1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))| z∈Δw (r)

13

μϕ (D(w, r)) A(D(w, r))

1 sup(1 − |z|2 ) (1 − |w|2 ) |Kw (ϕ(z))| . ε z∈D

(64)

Next, if (59) holds then by the Cauchy–Schwarz inequality, C ≤ sup z∈D

≤ and the conclusion follows by Theorem 4.2.

| < Cϕ Kw , Kz > | ||Kw || ||Kz ||

||Cϕ Kw || , ||Kw || 2

The Nevanlinna counting function for ϕ is defined for w ∈ D by  Nϕ (w) = (1 − |z|2 ) , ϕ(z)=w

where it is understood that if w is not a value of ϕ then Nϕ (w) = 0. By Littlewood’s inequality [24, page 187], Nϕ (w) it is easy to see that 1−|w| 2 dA(w) is a Carleson measure. Proposition 4.3. Let ϕ be a non-constant analytic self-map of D. Then for each ε > 0, ϕ is boundedly valent on Λε ∩ {|z| > 1/2}. Proof. Fix ε > 0 and recall that Λε = {z ∈ D : 1 − |z|2 > ε (1 − |ϕ(z)|2 )}. By [24, Section 10.4] it is easy to see that if w = ϕ(z) with z ∈ Λε and |z| > 1/2 then ε (1 − |w|2 )ηϕ,ε (w) ≤ Nϕ (w) ≤ C (1 − |w|2 ) ,

(65)

where ηϕ,ε (w) denotes the cardinality of ϕ−1 (w) ∩ Λε counting multiplicities. Thus ϕ is boundedly valent on Λε ∩ {|z| > 1/2} and the result is proved. 2 The Nevanlinna counting function for A2 is defined for w ∈ D by  N2,ϕ (w) = (1 − |z|2 )2 ,

(66)

ϕ(z)=w

where it is understood that if w is not a value of ϕ then N2,ϕ (w) = 0. By [23, Section 6.3], it is easy to see N2,ϕ (w) that (1−|w| 2 )2 dA(w) is a Carleson measure. The authors of [9, Theorem 2] launched the study of closed range composition operators on H 2 by using the pull-back measure of Lebesgue measure on T. Below is the main result of this section. We provide an analog of their result, on A2 , where we use the pull-back measure of normalized area measure on D. Not all equivalences below are new. In particular, (f) ⇔ (g) was proved in [2, Theorem 2.4] and (a) ⇔ (g) is the case for p = 2 and α = 2 in [25, Theorem 5.4]. Notice that while condition (f) requires Gε to intersect each D(w, r) at an area that is comparable to the area of D(w, r), by condition (b) this is equivalent to each D(w, r) merely intersecting Gε . Moreover notice that condition (e) makes it easy to check whether Cϕ is closed range on A2 . Remark 4.1. By Lemma 2.1 and Lemma 2.2 in [2], to prove that Cϕ is closed range on A2 we may assume that ϕ(0) = 0 and show that Cϕ is closed range on the invariant subspace of A2 of functions vanishing at the origin. This will be useful in the proof of (d) ⇒ (f) below, where we use the equivalent norm of A2 given in (15).

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.14 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

14

Theorem 4.4. The following statements are equivalent. N2,ϕ (ζ) (a) The Carleson measure (1−|ζ| 2 )2 dA(ζ) satisfies the reverse Carleson condition. (b) There exists ε > 0 such that Gε is hyperbolically dense. (c) The Carleson measure μϕ satisfies the reverse Carleson condition. (d) For all w ∈ D, ||Cϕ Kw ||  ||Kw ||. (e) There exists C > 0 such that for all w ∈ D C ≤ sup z∈D

(1 − |z|2 ) (1 − |w|2 ) . |1 − wϕ(z)|2

(f) There exists ε > 0 such that Gε = {ϕ(z) : (1 − |z|2 ) > ε (1 − |ϕ(z)|2 )} satisfies the reverse Carleson condition. (g) Cϕ is closed range on A2 . Proof. We will show that (a) ⇒ (b) ⇒ (c) ⇔ (e) ⇔ (d) ⇒ (f) ⇔ (g) ⇔ (a). We start by showing (a) ⇒ (b). By assumption we may choose 0 < r < 1 such that  N2,ϕ (ζ) dA(ζ)  A(D(w, r)) , w ∈ D . (1 − |ζ|2 )2

(67)

D(w,r)

Fix w ∈ D, recall (25) and let δw (r) = By (67) and since

Nϕ (ζ) 1−|ζ|2

1 − |z|2 . 2 z∈Δw (r) 1 − |ϕ(z)| sup

dA(ζ) is a Carleson measure it is easy to see that  A(D(w, r) ≤ C δw (r)

Nϕ (ζ) dA(ζ) 1 − |ζ|2

D(w,r)

≤ C δw (r) A(D(w, r)) . It follows that there exists an ε > 0 such that if w ∈ D then δw (r) ≥ ε. We conclude that there exists 0 < r < 1 such that for all w ∈ D, there exists zw ∈ Δw (r) ∩ Λε or equivalently that Gε is hyperbolically dense and (b) holds. Next we show that (b) ⇒ (c). This is an immediate consequence of Proposition 3.2 and Theorem 4.2. But we will provide below another proof that may be of independent interest. By our assumption we may choose ε > 0 such that Gε is hyperbolically dense. It follows that there exists 0 < r < 1 with the property that given w ∈ D there exists zw ∈ Δw (r) satisfying 1 − |zw | ≥ ε (1 − |w|) .

(68)



r + 12 1 . ϕ D(zw , ) ⊂ D w, 2 1 + 2r

(69)

We claim that

Indeed if ρ(z, zw ) < 1/2 then by the Invariant Schwarz Lemma, see [24, page 60], ϕ(z) ∈ D(ϕ(zw ), 1/2). Moreover since zw ∈ Δw (r), we have that w ∈ D(ϕ(zw ), r). Therefore by (11) we obtain ρ (ϕ(z), w) ≤

1 2

+r =: r 1 + 2r

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.15 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

15

and (69) holds. Thus by (12) and (68), for every w ∈ D and since zw ∈ Δw (r ) we obtain μϕ ((D(w, r ))) = A(Δw (r )) 1 ≥ A(D(zw , )) 2 1 ≥ (1 − |zw |2 )2 4 1 ≥ ε (1 − |w|2 )2 4  A(D(w, r )) .

(70)

We conclude that μϕ satisfies the reverse Carleson condition and (c) holds. By Theorem 4.2 and Theorem 4.3, (c), (d) and (e) are equivalent. Next we show that (d) ⇒ (f). If r ∈ (0, 1) and w ∈ D let  2 2 I(w, r) := (1 − |w| ) |(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z) . D\Δw (r)

Then by the Schwarz–Pick lemma,  I(w, r) = 4(1 − |w| )

2 2 D\Δw (r)

|w|2 (1 − |z|2 )2 |ϕ (z)|2 dA(z) |1 − wϕ(z)|6



≤ 16 (1 − |w|2 )2 D\Δw (r)

1 dA(z) |1 − wϕ(z)|4



= 16 (1 − |w| )

|Kw (ϕ(z))|2 dA(z)

2 2 D\Δw (r)



= 16 (1 − |w|2 )2

|Kw (z)|2 dμϕ (z) . D\D(w,r)

By Theorem 4.2 our hypothesis (d) is equivalent to μ ˜ϕ being bounded above and below on D. Then by the proof of Theorem 4.1 we see that (48) is valid. We conclude that lim sup I(w, r) = 0

r→1 w∈D

or equivalently that 

|(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z) = 0 .

lim sup (1 − |w|2 )2

r→1 w∈D

D\Δw (r)

Therefore there exists r ∈ (0, 1) such that for all w ∈ D  r) := (1 − |w|2 )2 I(w,



|(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z)

Δw (r)



 μ ˜ϕ (w) − (1 − |w| )

2 2 D\Δw (r)

|(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z)

(71)

JID:YJMAA 16

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.16 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

≥

1 μ ˜ϕ (w) ≥ C , 2

(72)

where (15) and (51) were used in the second line of the display above and the hypothesis (d), which is equivalent to μ˜ϕ being bounded below on D, was used in the third line of the display above. Next, if r ∈ (0, 1), ε > 0 and w ∈ D, let J(w, r, ε) denote the expression 

|(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z) .

(1 − |w|2 )2 Δw (r)\Λε ∩ϕ−1 {|ζ|>1/2}

Then,  J(w, r, ε) ≤ 4 ε (1 − |w| )

2 2 Δw (r)\Λε ∩ϕ−1 {|ζ|>1/2}

|ϕ (z)|2 (1 − |z|2 ) (1 − |ϕ(z)|2 ) dA(z) , |1 − wϕ(z)|6

and by making a non-univalent change of variables as in [24, 10.3] we obtain  J(w, r, ε) ≤ 4 ε (1 − |w| )

2 2 D(w,r)\Gε ∩{|ζ|>1/2}

(1 − |ζ|2 ) Nϕ (ζ) dA(ζ) . |1 − wζ|6

By (12), (13) and [24, 10.4] and for all w ∈ D we obtain 

(1 − |ζ|2 )2 dA(ζ) |1 − wζ|6

J(w, r, ε) ≤ C ε (1 − |w| )

2 2 D(w,r)

≤C ε.

(73)

 r, ε) denotes the Therefore, by (72) and (73), there exist r ∈ (0, 1) and ε > 0, such that if w ∈ D and J(w, expression 

|(Cϕ Kw ) (z)|2 (1 − |z|2 )2 dA(z)

(1 − |w| )

2 2 Δw (r)∩Λε ∩ϕ−1 {|ζ|>1/2}

then  r, ε) ≥ C − J(w, r, ε) ≥ C . J(w,

(74)

Therefore by making once again a non-univalent change of variables as in [24, 10.3] and by [23, 6.3] we conclude that if w ∈ D then  C ≤ (1 − |w| )

2 2 Δw (r)∩Λε ∩ϕ−1 {|ζ|>1/2}



= (1 − |w| )

2 2 D(w,r)∩Gε ∩{|ζ|>1/2}



≤ C (1 − |w| )

2 2 D(w,r)∩Gε

(1 − |z|2 )2 |ϕ (z)|2 dA(z) |1 − wϕ(z)|6

N2,ϕ (ζ) dA(ζ) |1 − wζ|6

(1 − |ζ|2 )2 dA(ζ) |1 − wζ|6

(75)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.17 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

17

and by (12), (13) C≤

A(D(w, r) ∩ Gε ) . A(D(w, r))

By Theorem 1 in [2], (f) and (g) are equivalent. Lastly by Theorem 5.4 in [25] (a) and (g) are equivalent. This completes the proof of the theorem. 2 Remark 4.2. The Bloch space B, is the Banach space of analytic functions on D such that ||f ||B = sup |f  (z)| (1 − |z|2 ) < ∞ . z∈D

For ε > 0, let |ϕ (z)| (1 − |z|2 ) > ε} 1 − |ϕ(z)|2

Λε = {z ∈ D :

and Fε = ϕ(Λε ). By [13] and [8] or [3], Cϕ is closed range on the Bloch space if and only if there exists and ε > 0 such that Fε is hyperbolically dense (this is condition (iii) in [3, Theorem 2.2]). Moreover in [3, Corollary 2.3] it was shown to be equivalent to ||Cϕ (αp )||B  1, for p ∈ D. The similarity of this with our results on closed range operators on A2 is interesting. 5. On Hardy spaces Let ϕ be an analytic self-map of D and recall from Section 4 that the Nevanlinna counting function, Nϕ (ζ), for H 2 is 

Nϕ (ζ) =

(1 − |z|2 ) , ζ ∈ D .

(76)

ϕ(z)=ζ

Let f ∈ H 2 . Then by (16) and by making a non-univalent change of variables as done in [24, page 186] we can easily see that  ||Cϕ f ||2H 2

 |f (ϕ(0))| + 2

|f  (ϕ(z)|2 |ϕ (z)|2 (1 − |z|2 ) dA(z)

D

 = |f (ϕ(0))|2 +

|f  (ζ)|2 Nϕ (ζ) dA(ζ) .

D

Therefore recalling that the reproducing kernel for H 2 is Kw (z) =

1 , 1 − wz

we obtain for w ∈ D that  ||Cϕ Kw ||2H 2  |Kw (ϕ(0))|2 +

 |Kw (ζ|2 Nϕ (ζ)dA(ζ) .

(77)

D

Below is the first main theorem of this section. The result is not new. In particular the equivalence (a) ⇔ (b) follows by [25, Theorem 5.4]. Moreover Luery in her thesis [18] proved that (b) ⇔ (c) using the Aleksandrov– Clark measures. Our new short proof below uses pseudohyperbolic disks. We may assume below that

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.18 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

18

ϕ(0) = 0, and prove the result for the invariant subspace of H 2 of functions vanishing at the origin, see for example Remark 3.1 in [25]. Theorem 5.1. Let ϕ be an analytic self-map of D. The following statements are equivalent. Nϕ (ζ) (a) The Carleson measure 1−|ζ| 2 dA(ζ) satisfies the reverse Carleson condition. (b) Cϕ is closed range on H 2 . (c) For all w ∈ D, ||Cϕ Kw ||  ||Kw ||. Proof. By [25, Theorem 5.4] we have (a) ⇔ (b). The implication (b) ⇒ (c) is trivial. Lastly we need to show that (c) ⇒ (a). Assume that (c) holds. Then by (77) 

 |Kw (ζ)|2 Nϕ (ζ) dA(ζ) 

D

|w|2 1 − |w|2

or  γw := (1 − |w|2 ) D

By Proposition 4.1, for the Carleson measure

Nϕ (ζ) 1−|ζ|2

 lim sup

r→1 w∈D

1 Nϕ (ζ) dA(ζ)  1 . |1 − w ζ|4

D\D(w,r)

dA(ζ) with α = β = 1, we obtain

1 − |w|2 Nϕ (ζ) dA(ζ) = 0 . |1 − wζ|4

(78)

Therefore, if ε = (1/2) inf w∈D γw > 0, there exists δ > 0 such that if r > 1 − δ, and w ∈ D, 

1 Nϕ (ζ) dA(ζ) |1 − wζ|4

C ≥ (1 − |w|2 ) D(w,r)

⎛



⎜ = (1 − |w|2 ) ⎝ 

D

= γw − D\D(w,r)

1 Nϕ (ζ) dA(ζ) − |1 − wζ|4

 D\D(w,r)

⎞ 1 ⎟ Nϕ (ζ) dA(ζ)⎠ |1 − wζ|4

1 − |w|2 Nϕ (ζ) dA(ζ) |1 − wζ|4

≥ γw − ε = γw − (1/2) inf γw w∈D

≥ (1/2) inf γw > 0 . w∈D

By (12) and (13) we conclude that if w ∈ D then 

1 1 (1 − |w|2 )2

Nϕ (ζ) dA(ζ) 1 − |ζ|2

D(w,r)



1 A(D(w, r))



Nϕ (ζ) dA(ζ) . 1 − |ζ|2

D(w,r)

This shows that

Nϕ (ζ) 1−|ζ|2

dA(ζ) satisfies the reverse Carleson condition and (a) is proved. 2

(79)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.19 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

19

It is a consequence of the Schwarz–Pick Lemma that, for all z ∈ D, 1 − |z|2 ≤ C (1 − |ϕ(z)|2 ). Recalling the definitions of the Nevanlinna counting functions for A2 and H 2 given in (66) and (76), if ζ ∈ D then Nϕ (ζ) N2,ϕ (ζ) ≤C . 2 2 (1 − |ζ| ) 1 − |ζ|2 Now the following is an immediate corollary of Theorems 4.4 and 5.1. It was first proved by Zorboska in [27, Corollary 4.2]. Corollary 5.1. If Cϕ is closed range on A2 then it is closed range on H 2 . By [27, Corollary 4.3] the converse of the above holds, if ϕ is univalent. We next provide a more general condition that guarantees this. Recall that ηϕ (w) denotes the cardinality of ϕ−1 (w) counting multiplicities. Corollary 5.2. Suppose that ηϕ (ζ) dA(ζ) is a Carleson measure. Then Cϕ is closed range on H 2 if and only if Cϕ is closed range on A2 . N (ζ)

ϕ Proof. First, assume that Cϕ is closed range on H 2 . By Theorem 5.1, 1−|ζ| 2 dA(ζ) satisfies the reverse Carleson condition. Therefore we may choose 0 < r < 1 such that for each w ∈ D,



Nϕ (ζ) dA(ζ)  |D(w, r)| . 1 − |ζ|2

(80)

D(w,r)

Given w ∈ D, recall (25) and let δw (r) =

1 − |z|2 . 2 z∈Δw (r) 1 − |ϕ(z)| sup

By (80) and since by assumption ηϕ (ζ) dA(ζ) is a Carleson measure, we obtain  A(D(w, r)) ≤ C δw (r)

ηϕ (ζ) dA(ζ) D(w,r)

≤ C δw (r) A(D(w, r))

(81)

and therefore, there exists ε > 0 such that for all w ∈ D, δw (r) ≥ ε. We conclude that Gε is hyperbolically dense and, by Theorem 4.4, Cϕ is closed range on A2 . The other direction follows by Corollary 5.1. 2 An analytic function f on D is said to belong to the Hardy space H 1 if 1 sup 0
2π |f (reiθ )| dθ < ∞ . 0

Let ϕ be an analytic self-map of D. The pull-back measure of the Lebesgue measure m on the unit circle T under the map ϕ is defined on E, a Borel subset of T, by mϕ (E) = m(ϕ−1 (E)) .

JID:YJMAA 20

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.20 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

By [22, page 50] (or [20]), mϕ is a measure that is absolutely continuous with respect to Lebesgue measure m on T and its Radon–Nikodym derivative is bounded. Moreover if E is a Borel subset of T then  mϕ (E) ≤ E

1 − |ϕ(0)|2 dt . |1 − ϕ(0)eit |2

Proposition 5.1. Suppose ϕ is a non-constant analytic self-map of D such that T ⊆ ϕ(T). If ϕ ∈ H 1 , then Cϕ is closed range on H 2 . Proof. By hypothesis T ⊆ ϕ(T), hence if F is a measurable subset of T, then F = ϕ(E) where E = ϕ−1 (F ). Now if En is a sequence of measurable subsets of T such that m(En ) → 0 as n → ∞ and since ϕ ∈ H 1 ,  lim

n→∞ En

|ϕ (t)| dt = 0 .

Equivalently if mϕ (ϕ(En )) → 0 then m(ϕ(En )) → 0. Therefore the Lebesque measure m is absolutely continuous with respect to mϕ . By [9, Theorem 2] Cϕ is closed range on H 2 . 2 By [2, Theorem 2.5], the only univalent self-maps of D that induce a closed range composition operator Cϕ on A2 and, by [27, Corollary 4.2], on H 2 are the conformal automorphisms of D. Two examples of non-inner functions that give rise to closed range composition operators on H 2 are given in [9, page 218]. Moreover all known examples of closed range composition operators on A2 involve inner functions. So it is natural to ask if there are any such examples with non-inner functions. Next we provide an example of a boundedly valent outer function ϕ such that Cϕ is closed range on H 2 and on A2 . The main idea regarding its construction was developed in a conversation with J. Akeroyd. We are grateful to him for allowing us to include it here. If an analytic function f maps D conformally onto a bounded domain G then by Caratheodory’s theorem (see [21, Theorem 2.6]) f has a continuous injective extension to D. Moreover by [21, Proposition 6.2, Theorem 6.8] if f (T) is a rectifiable curve then f  ∈ H 1 . Let G denote the half annulus defined by 

 w :

1 < |w| < 1 , Im(w) > 0 2

,

F the Riemann mapping of D onto G and ϕ = F 3 . Then ϕ is three-valent, ϕ(T) ⊇ T and by the discussion above it is easy to see that F  ∈ H 1 and therefore ϕ ∈ H 1 . Moreover since |ϕ(z)| > 1/2 for all z ∈ D, ϕ is an outer function, see for example [10, Remark 3.2.5]. By Proposition 5.1 Cϕ is closed range on H 2 which, by Corollary 5.2 is equivalent to Cϕ being closed range on A2 . 6. On Dirichlet spaces In this section we will assume that Cϕ is a bounded operator on D. It is well known that this is equivalent to the measure ηϕ (w) dA(w) being a Carleson measure on D, see for example [5,19,14]. Moreover if ϕ is an analytic self-map of D with ϕ(0) = 0, then Cϕ is closed range on D if and only if Cϕ is closed range on D0 = {f ∈ D : f (0) = 0}. If ϕ(0) = 0, letting ψ = αϕ(0) ◦ ϕ, then Cϕ is closed range on D if and only if Cψ is closed range on D0 . Therefore in this section we shall assume that ϕ(0) = 0 and consider Cϕ on D0 . Moreover, for a non-constant symbol ϕ the composition operator Cϕ is injective; therefore it is closed range if and only if it is bounded below.

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.21 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

21

Recall the notation introduced in Section 3, where for ε > 0 we let  Λε =

1 1 z ∈ D : log > ε2 log 1 − |ϕ(z)|2 1 − |z|2

 (82)

and Gε = ϕ(Λe ). ϕ (ζ) for ζ ∈ D by Define the Nevanlinna type counting function N 

ϕ (ζ) = N

ϕ(z)=ζ

1 log 1 − |z|2

−1 ,

(83)

ϕ (ζ) = 0. By Schwarz’s Lemma if z, ζ ∈ D are such where it is understood that if ζ is not in ϕ(D) then N that ϕ(z) = ζ then 1 log 1 − |ζ|2



1 log 1 − |z|2

−1 ≤1

and therefore log

1 ϕ (ζ) ≤ ηϕ (ζ) . N 1 − |ζ|2

(84)

1  Since ηϕ (ζ) dA(ζ) is a Carleson measure it is clear that the measure log 1−|ζ| 2 Nϕ (ζ) dA(ζ) is also a Carleson measure. If g is a non-negative measurable function on D then by making a non-univalent change of variables similar to [23, page 398] we obtain





2

g(ϕ(z))|ϕ (z)|

1 log 1−|ϕ(z)| 2

log

D

1 1−|z|2

 dA(z) =

g(ζ) log D

1 ϕ (ζ)dA(ζ). N 1 − |ζ|2

(85)

Proposition 6.1. The following three statements are equivalent: 1  (a) The measure log 1−|ζ| 2 Nϕ (ζ) dA(ζ) satisfies the reverse Carleson condition. 1  (b) There exists an ε > 0 such that the measure χGε (ζ) log 1−|ζ| 2 Nϕ (ζ) dA(ζ) satisfies the reverse Carleson condition. (c) There exists an ε > 0 such that the measure χGε (ζ)ηϕ (ζ)dA(ζ) satisfies the reverse Carleson condition. Proof. First, we assume (a). We may then choose r ∈ (0, 1) so that for each w ∈ D,  log

1 ϕ (ζ) dA(ζ)  A(D(w, r)) . N 1 − |ζ|2

D(w,r)

Moreover for each ε > 0, w ∈ D, (84) and since ηϕ (ζ) dA(ζ) is a Carleson measure,  log D(w,r)\Gε

1 ϕ (ζ) dA(ζ) ≤ ε2 N 1 − |ζ|2

 ηϕ (ζ) dA(ζ)

D(w,r)

≤ ε2 C A(D(w, r)) ;

(86)

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.22 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

22

therefore by (86) we may choose ε > 0 small enough such that  1 ϕ (ζ) dA(ζ)  A(D(w, r)) log N 1 − |ζ|2 D(w,r)∩Gε

and (b) follows. By (84), it is immediate that (b) ⇒ (c). Lastly assume that (c) holds, and let r ∈ (0, 1) be such that for all w ∈ D,  ηϕ (ζ) dA(ζ)  A(D(w, r)) . D(w,r)∩Gε

Then, by (25) and (85), for each w ∈ D 



1 ϕ (ζ) dA(ζ) ≥ log N 1 − |ζ|2

D(w,r)

log

1 ϕ (ζ) dA(ζ) N 1 − |ζ|2

D(w,r)∩Gε



1 log 1−|ϕ(z)| 2

≥ Δw (r)∩Λε

1 log 1−|z| 2

|ϕ (z)|2 dA(z)



≥ε

2

ηϕ (ζ) dA(ζ) D(w,r)∩Gε

 A(D(w, r)) and (a) follows. 2 The following is an immediate corollary of Theorem 4.1. Corollary 6.1. Each of the following three statements is equivalent to (a), (b), (c) of the proposition above. (1)  inf

w∈D D

(1 − |w|2 )2 1 ϕ (ζ) dA(ζ) ≥ C . N log 4 |1 − wζ| 1 − |ζ|2

(2) There exists ε > 0 such that  inf

w∈D Gε

1 (1 − |w|2 )2 ϕ (ζ) dA(ζ) ≥ C . log N 4 |1 − wζ| 1 − |ζ|2

(3) There exists ε > 0 such that  inf

w∈D Gε

(1 − |w|2 )2 ηϕ (ζ) dA(ζ) ≥ C . |1 − wζ|4

Proposition 6.2. The following three statements are equivalent: (A) For all f ∈ D,   1 ϕ (ζ) dA(ζ) . |f  (z)|2 dA(z) ≤ C |f  (ζ)|2 log N 1 − |ζ|2 D

D

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.23 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

(B) There exists ε > 0 such that for all f ∈ D,    2 |f (z)| dA(z) ≤ C |f  (ζ)|2 log D

Gε

23

1 ϕ (ζ) dA(ζ) . N 1 − |ζ|2

(C) There exists ε > 0 such that for all f ∈ D,    2 |f (z)| dA(z) ≤ C |f  (ζ)|2 ηϕ (ζ) dA(ζ) . D

Gε

Proof. First assume that (A) holds. If (B) is not valid then we can find a sequence fn ∈ D with fn (0) = 0 and ||fn ||D = 1 for all n, and  1 ϕ (ζ) dA(ζ) = 0 . N lim |fn (ζ)|2 log (87) n→∞ 1 − |ζ|2 G1

n

Note that by (82) and (85) 

|fn (ζ)|2

D\G 1

1 ϕ (ζ) dA(ζ) ≤ log N 1 − |ζ|2

n





|(fn ◦ ϕ) (z)|

2

D\Λ 1

n

1 ≤ 2 n



1 log 1−|ϕ(z)| 2 1 log 1−|z| 2

|ϕ (z)|2 dA(z)

|(fn ◦ ϕ) (z)|2 dA(z)

D\Λ 1

n

1 ≤ 2. n

(88)

By (87) and (88) we conclude that  lim

n→∞

|fn (ζ)|2 log

D

1 ϕ (ζ) dA(ζ) = 0 N 1 − |ζ|2

and thusly (A) is not valid. We have shown that (A) ⇒ (B). By (84), the implication (B) ⇒ (C) is clear. Finally, if (C) holds, then by (87) and (88) and for each f ∈ D, 

|f  (ζ)|2 log

D

1 ϕ (ζ) dA(ζ) ≥ N 1 − |ζ|2



|f  (ζ)|2 log

Gε

 ≥ Λε

1 ϕ (ζ) dA(ζ) N 1 − |ζ|2

|(f ◦ ϕ) (z)|2 

≥ ε2 Gε

1 log 1−|ϕ(z)| 2 1 log 1−|z| 2

|f  (ζ)|2 ηϕ (ζ) dA(ζ) 

≥Cε

2

|f  (ζ)|2 dA(ζ)

D

and (A) follows. 2 Proposition 3.2 is an essential ingredient of the following corollary.

|ϕ (z)|2 dA(z)

JID:YJMAA 24

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.24 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

1  Corollary 6.2. Assume that the measure log 1−|ζ| 2 Nϕ (ζ)dA(ζ) satisfies the reverse Carleson condition. Then the following hold. (a) There exists an ε > 0 such that Gε is hyperbolically dense. (b) There exists r ∈ (0, 1) such that for all w ∈ D,

A(ϕ−1 (D(w, r))) ≥ C A(D(w, r))1/ε . 2

(c) For every w ∈ D, ||Cϕ Kw ||D  ||Kw ||D . Proof. For each w ∈ D and r ∈ (0, 1) we define

δw (r) =

sup

log

z∈Δw (r)

1 1 − |ϕ(z)|2

log

1 1 − |z|2

−1 .

By our assumption and since ηϕ (ζ)dA(ζ) is a Carleson measure, there exists r ∈ (0, 1) such that for all w ∈ D,  A(D(w, r)) ≤

log

1 ϕ (ζ) dA(ζ) N 1 − |ζ|2

D(w,r)

 ≤ δw (r)

ηϕ (ζ) dA(ζ) D(w,r)

≤ C δw (r) A(D(w, r)) ,

(89)

and δw (r) ≥ C. This is equivalent to Gε being hyperbolically dense for some ε > 0 and (a) follows. Next, by (a) we know that Gε is hyperbolically dense. Therefore there exists r ∈ (0, 1) such that for all w ∈ D there exists zw ∈ Δw (r) ∩ Λε . By (69) and (82) we obtain A(ϕ−1 (D(w, r))) ≥ C (1 − |zw |2 )2 ≥ C (1 − |ϕ(zw )|2 )2/ε

2

1/ε2

≥ C A(D(w, r))

and (b) follows. Lastly, (c) is an immediate consequence of (a) and Proposition 3.2. 2 Let (a), (b), (c), (A), (B), (C) be the statements in Proposition 6.1 and in Proposition 6.2. Moreover recall the three statements in Corollary 6.1 which are equivalent to (a), (b) and (c). An immediate consequence of [16, Theorem 4.3] is that (A) ⇒ (a), (B) ⇒ (b) and (C) ⇒ (c). Consider the following three statements: (e) There exists an ε > 0 such that Gε is hyperbolically dense. (E) For all w ∈ D, ||Cϕ Kw ||D  ||Kw ||D . (CR) The operator Cϕ is closed range on D.

JID:YJMAA

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.25 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

25

With the diagram below we summarize the results in this section. A

B

C

a

b

c

e

E

Jovovic and MacCluer showed in [14] that if Cϕ is closed range in D then ηϕ (ζ) dA(ζ) satisfies the reverse Carleson condition. Moreover, Luecking showed in [17] that the converse is false. Therefore by Theorem 4.1 the condition  (1 − |w|2 )2 inf ηϕ (ζ) dA(ζ) ≥ C w∈D |1 − wζ|4 D

is a necessary but not a sufficient condition for Cϕ to be closed range on D. It is clear that each of (A), (B), (C) imply (CR) and that (CR) implies (E). We conjecture the following. Conjecture. Cϕ is closed range in D if and only if there exists ε > 0 such that the measure χGε (ζ) ηϕ (ζ) dA(ζ) satisfies the reverse Carleson condition, equivalently  inf

w∈D Gε

(1 − |w|2 )2 ηϕ (ζ) dA(ζ) ≥ C , |1 − wζ|4

that is (CR) ⇔ (c). Acknowledgment We wish to thank the referee for the time spent studying our manuscript and for his/her suggestions which helped improve the exposition of the manuscript. References [1] J. Agler, J.E. McCarthy, Pick Interpolation and Hilbert Function Spaces, Grad. Stud. Math., vol. 44, American Mathematical Society, Providence, RI, 2002. [2] J.R. Akeroyd, P. Ghatage, Closed-range composition operators on A2 , Illinois J. Math. 52 (2) (2008) 533–549. [3] J.R. Akeroyd, P. Ghatage, M. Tjani, Closed-range composition operators on A2 and the Bloch space, Integral Equations Operator Theory 68 (4) (2010) 503–517. [4] J. Akeroyd, S. Fulmer, Closed-range composition operators on weighted Bergman spaces, Integral Equations Operator Theory 72 (4) (2012) 103–114. [5] J. Arazy, S.D. Fisher, J. Peetre, Mobius invariant function spaces, J. Reine Angew. Math. 363 (1985) 110–145. [6] S. Axler, Bergman spaces and their operators, Pitman Res. Notes 171 (1988) 1–50. [7] S. Axler, D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (2) (1998) 387–400. [8] H. Chen, Boundedness from below of composition operators on the Bloch spaces, Sci. China Ser. A 46 (2003) 838–846 (English summary). [9] J. Cima, J. Thomson, W. Wogen, On some properties of composition operators, Indiana Univ. Math. J. 24 (1974/1975) 215–220. [10] J.A. Cima, W.T. Ross, The Backward Shift on the Hardy Space, Math. Surveys Monogr., vol. 79, American Mathematical Society, Providence, RI, 2000. [11] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995. [12] P. Duren, A. Schuster, Bergman Spaces, Math. Surveys Monogr., vol. 100, American Mathematical Society, Providence, RI, 2004.

JID:YJMAA 26

AID:19562 /FLA

Doctopic: Functional Analysis

[m3L; v1.157; Prn:15/06/2015; 13:12] P.26 (1-26)

P. Ghatage, M. Tjani / J. Math. Anal. Appl. ••• (••••) •••–•••

[13] P. Ghatage, D. Zheng, N. Zorboska, Sampling sets and closed range composition operators on the Bloch space, Proc. Amer. Math. Soc. 133 (5) (2005) 1371–1377. [14] M. Jovovic, B.D. MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1–2) (1997) 229–247. [15] D.H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981) 1–11. [16] D.H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985) 85–111. [17] D.H. Luecking, Bounded composition operators with closed range on the Dirichlet space, Proc. Amer. Math. Soc. 128 (4) (2000) 1109–1116. [18] K. Luery, Composition operators on Hardy spaces of the disk and half-space, Doctoral Dissertation, University of Florida, 2013. [19] B.D. MacCluer, J.H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (4) (1986) 878–906. [20] E.A. Nordgren, Composition operators, Canad. J. Math. 20 (1968) 442–449. [21] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, New York, 1991. [22] J.H. Schwartz, Composition operators on H p , Doctoral Dissertation, University of Toledo, 1969. [23] J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1) (1987) 375–404. [24] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. [25] M. Tjani, Closed range composition operators on Besov type space, Complex Anal. Oper. Theory 8 (1) (2014) 189–212. [26] K. Zhu, Operator Theory in Function Spaces, second edition, Math. Surveys Monogr., vol. 138, American Mathematical Society, Providence, RI, 2007. [27] N. Zorboska, Composition operators with closed range, Trans. Amer. Math. Soc. 344 (2) (1994) 791–801.